SlideShare a Scribd company logo
If D is dense in a topological space (X, tau), prove that D is dense in every topology smaller
than tau.
Solution
If (X,T) is atopological space, then X is the set and T is the topology on X.T is also the family of
subsets of X.
If an element in the set X, D refers to 'dense', then, an element D in T refers to dense only. A
subset can derive elements only from its superset.

More Related Content

More from arsmobiles

Define nonconcordant variation Why does nonconcordant variation mak.pdf
Define nonconcordant variation Why does nonconcordant variation mak.pdfDefine nonconcordant variation Why does nonconcordant variation mak.pdf
Define nonconcordant variation Why does nonconcordant variation mak.pdf
arsmobiles
 
Describe the operation of a tubular bowl centrifugeSolutionTub.pdf
Describe the operation of a tubular bowl centrifugeSolutionTub.pdfDescribe the operation of a tubular bowl centrifugeSolutionTub.pdf
Describe the operation of a tubular bowl centrifugeSolutionTub.pdf
arsmobiles
 
Could you help me with this question Thanks so much!Let T be an e.pdf
Could you help me with this question Thanks so much!Let T be an e.pdfCould you help me with this question Thanks so much!Let T be an e.pdf
Could you help me with this question Thanks so much!Let T be an e.pdf
arsmobiles
 
Briefly discuss Philippa Foot’s contributions to virtue theory.S.pdf
Briefly discuss Philippa Foot’s contributions to virtue theory.S.pdfBriefly discuss Philippa Foot’s contributions to virtue theory.S.pdf
Briefly discuss Philippa Foot’s contributions to virtue theory.S.pdf
arsmobiles
 
List for a company you either work or are familiar with, the interna.pdf
List for a company you either work or are familiar with, the interna.pdfList for a company you either work or are familiar with, the interna.pdf
List for a company you either work or are familiar with, the interna.pdf
arsmobiles
 
Write a C program that uses a recursive function to print a triangle .pdf
Write a C program that uses a recursive function to print a triangle .pdfWrite a C program that uses a recursive function to print a triangle .pdf
Write a C program that uses a recursive function to print a triangle .pdf
arsmobiles
 
Which avenue of research is least likely to reveal whether asexual or.pdf
Which avenue of research is least likely to reveal whether asexual or.pdfWhich avenue of research is least likely to reveal whether asexual or.pdf
Which avenue of research is least likely to reveal whether asexual or.pdf
arsmobiles
 
When testing the difference between two population means, the varian.pdf
When testing the difference between two population means, the varian.pdfWhen testing the difference between two population means, the varian.pdf
When testing the difference between two population means, the varian.pdf
arsmobiles
 
IT Project Management1. Please compare and contrast a traditional .pdf
IT Project Management1. Please compare and contrast a traditional .pdfIT Project Management1. Please compare and contrast a traditional .pdf
IT Project Management1. Please compare and contrast a traditional .pdf
arsmobiles
 
Whats the relationship between DNA and RNA Whats the relat.pdf
Whats the relationship between DNA and RNA Whats the relat.pdfWhats the relationship between DNA and RNA Whats the relat.pdf
Whats the relationship between DNA and RNA Whats the relat.pdf
arsmobiles
 
What does it mean to destroy something at the molecular level in ter.pdf
What does it mean to destroy something at the molecular level in ter.pdfWhat does it mean to destroy something at the molecular level in ter.pdf
What does it mean to destroy something at the molecular level in ter.pdf
arsmobiles
 
What is a rain shadow, and why does it happenSolutionThe warm.pdf
What is a rain shadow, and why does it happenSolutionThe warm.pdfWhat is a rain shadow, and why does it happenSolutionThe warm.pdf
What is a rain shadow, and why does it happenSolutionThe warm.pdf
arsmobiles
 
Use transformations to graph this function. (Do this on paper. Your t.pdf
Use transformations to graph this function. (Do this on paper. Your t.pdfUse transformations to graph this function. (Do this on paper. Your t.pdf
Use transformations to graph this function. (Do this on paper. Your t.pdf
arsmobiles
 
Understand how radioactive isotopes are used in dating fossilsSo.pdf
Understand how radioactive isotopes are used in dating fossilsSo.pdfUnderstand how radioactive isotopes are used in dating fossilsSo.pdf
Understand how radioactive isotopes are used in dating fossilsSo.pdf
arsmobiles
 
umbrella term for all healthcare plans that focus on reducing the cos.pdf
umbrella term for all healthcare plans that focus on reducing the cos.pdfumbrella term for all healthcare plans that focus on reducing the cos.pdf
umbrella term for all healthcare plans that focus on reducing the cos.pdf
arsmobiles
 
) Code the following class. Hopefully the method names clearly imply.pdf
) Code the following class. Hopefully the method names clearly imply.pdf) Code the following class. Hopefully the method names clearly imply.pdf
) Code the following class. Hopefully the method names clearly imply.pdf
arsmobiles
 
The rate of diffusion (J) depends on several physical factors. Brief.pdf
The rate of diffusion (J) depends on several physical factors. Brief.pdfThe rate of diffusion (J) depends on several physical factors. Brief.pdf
The rate of diffusion (J) depends on several physical factors. Brief.pdf
arsmobiles
 
The mutations that make HIV resistant to specific drugs are usually..pdf
The mutations that make HIV resistant to specific drugs are usually..pdfThe mutations that make HIV resistant to specific drugs are usually..pdf
The mutations that make HIV resistant to specific drugs are usually..pdf
arsmobiles
 
The low density of ice makes it ineffective at insulating water from .pdf
The low density of ice makes it ineffective at insulating water from .pdfThe low density of ice makes it ineffective at insulating water from .pdf
The low density of ice makes it ineffective at insulating water from .pdf
arsmobiles
 
The graph represents a correspondence between x and y. Determine whet.pdf
The graph represents a correspondence between x and y. Determine whet.pdfThe graph represents a correspondence between x and y. Determine whet.pdf
The graph represents a correspondence between x and y. Determine whet.pdf
arsmobiles
 

More from arsmobiles (20)

Define nonconcordant variation Why does nonconcordant variation mak.pdf
Define nonconcordant variation Why does nonconcordant variation mak.pdfDefine nonconcordant variation Why does nonconcordant variation mak.pdf
Define nonconcordant variation Why does nonconcordant variation mak.pdf
 
Describe the operation of a tubular bowl centrifugeSolutionTub.pdf
Describe the operation of a tubular bowl centrifugeSolutionTub.pdfDescribe the operation of a tubular bowl centrifugeSolutionTub.pdf
Describe the operation of a tubular bowl centrifugeSolutionTub.pdf
 
Could you help me with this question Thanks so much!Let T be an e.pdf
Could you help me with this question Thanks so much!Let T be an e.pdfCould you help me with this question Thanks so much!Let T be an e.pdf
Could you help me with this question Thanks so much!Let T be an e.pdf
 
Briefly discuss Philippa Foot’s contributions to virtue theory.S.pdf
Briefly discuss Philippa Foot’s contributions to virtue theory.S.pdfBriefly discuss Philippa Foot’s contributions to virtue theory.S.pdf
Briefly discuss Philippa Foot’s contributions to virtue theory.S.pdf
 
List for a company you either work or are familiar with, the interna.pdf
List for a company you either work or are familiar with, the interna.pdfList for a company you either work or are familiar with, the interna.pdf
List for a company you either work or are familiar with, the interna.pdf
 
Write a C program that uses a recursive function to print a triangle .pdf
Write a C program that uses a recursive function to print a triangle .pdfWrite a C program that uses a recursive function to print a triangle .pdf
Write a C program that uses a recursive function to print a triangle .pdf
 
Which avenue of research is least likely to reveal whether asexual or.pdf
Which avenue of research is least likely to reveal whether asexual or.pdfWhich avenue of research is least likely to reveal whether asexual or.pdf
Which avenue of research is least likely to reveal whether asexual or.pdf
 
When testing the difference between two population means, the varian.pdf
When testing the difference between two population means, the varian.pdfWhen testing the difference between two population means, the varian.pdf
When testing the difference between two population means, the varian.pdf
 
IT Project Management1. Please compare and contrast a traditional .pdf
IT Project Management1. Please compare and contrast a traditional .pdfIT Project Management1. Please compare and contrast a traditional .pdf
IT Project Management1. Please compare and contrast a traditional .pdf
 
Whats the relationship between DNA and RNA Whats the relat.pdf
Whats the relationship between DNA and RNA Whats the relat.pdfWhats the relationship between DNA and RNA Whats the relat.pdf
Whats the relationship between DNA and RNA Whats the relat.pdf
 
What does it mean to destroy something at the molecular level in ter.pdf
What does it mean to destroy something at the molecular level in ter.pdfWhat does it mean to destroy something at the molecular level in ter.pdf
What does it mean to destroy something at the molecular level in ter.pdf
 
What is a rain shadow, and why does it happenSolutionThe warm.pdf
What is a rain shadow, and why does it happenSolutionThe warm.pdfWhat is a rain shadow, and why does it happenSolutionThe warm.pdf
What is a rain shadow, and why does it happenSolutionThe warm.pdf
 
Use transformations to graph this function. (Do this on paper. Your t.pdf
Use transformations to graph this function. (Do this on paper. Your t.pdfUse transformations to graph this function. (Do this on paper. Your t.pdf
Use transformations to graph this function. (Do this on paper. Your t.pdf
 
Understand how radioactive isotopes are used in dating fossilsSo.pdf
Understand how radioactive isotopes are used in dating fossilsSo.pdfUnderstand how radioactive isotopes are used in dating fossilsSo.pdf
Understand how radioactive isotopes are used in dating fossilsSo.pdf
 
umbrella term for all healthcare plans that focus on reducing the cos.pdf
umbrella term for all healthcare plans that focus on reducing the cos.pdfumbrella term for all healthcare plans that focus on reducing the cos.pdf
umbrella term for all healthcare plans that focus on reducing the cos.pdf
 
) Code the following class. Hopefully the method names clearly imply.pdf
) Code the following class. Hopefully the method names clearly imply.pdf) Code the following class. Hopefully the method names clearly imply.pdf
) Code the following class. Hopefully the method names clearly imply.pdf
 
The rate of diffusion (J) depends on several physical factors. Brief.pdf
The rate of diffusion (J) depends on several physical factors. Brief.pdfThe rate of diffusion (J) depends on several physical factors. Brief.pdf
The rate of diffusion (J) depends on several physical factors. Brief.pdf
 
The mutations that make HIV resistant to specific drugs are usually..pdf
The mutations that make HIV resistant to specific drugs are usually..pdfThe mutations that make HIV resistant to specific drugs are usually..pdf
The mutations that make HIV resistant to specific drugs are usually..pdf
 
The low density of ice makes it ineffective at insulating water from .pdf
The low density of ice makes it ineffective at insulating water from .pdfThe low density of ice makes it ineffective at insulating water from .pdf
The low density of ice makes it ineffective at insulating water from .pdf
 
The graph represents a correspondence between x and y. Determine whet.pdf
The graph represents a correspondence between x and y. Determine whet.pdfThe graph represents a correspondence between x and y. Determine whet.pdf
The graph represents a correspondence between x and y. Determine whet.pdf
 

Recently uploaded

C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx
C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptxC1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx
C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx
mulvey2
 
How to Add Chatter in the odoo 17 ERP Module
How to Add Chatter in the odoo 17 ERP ModuleHow to Add Chatter in the odoo 17 ERP Module
How to Add Chatter in the odoo 17 ERP Module
Celine George
 
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
Nguyen Thanh Tu Collection
 
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptxChapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Mohd Adib Abd Muin, Senior Lecturer at Universiti Utara Malaysia
 
A Survey of Techniques for Maximizing LLM Performance.pptx
A Survey of Techniques for Maximizing LLM Performance.pptxA Survey of Techniques for Maximizing LLM Performance.pptx
A Survey of Techniques for Maximizing LLM Performance.pptx
thanhdowork
 
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Dr. Vinod Kumar Kanvaria
 
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRMHow to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
Celine George
 
Film vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movieFilm vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movie
Nicholas Montgomery
 
The basics of sentences session 6pptx.pptx
The basics of sentences session 6pptx.pptxThe basics of sentences session 6pptx.pptx
The basics of sentences session 6pptx.pptx
heathfieldcps1
 
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
RitikBhardwaj56
 
writing about opinions about Australia the movie
writing about opinions about Australia the moviewriting about opinions about Australia the movie
writing about opinions about Australia the movie
Nicholas Montgomery
 
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityNatural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Akanksha trivedi rama nursing college kanpur.
 
Main Java[All of the Base Concepts}.docx
Main Java[All of the Base Concepts}.docxMain Java[All of the Base Concepts}.docx
Main Java[All of the Base Concepts}.docx
adhitya5119
 
The Diamonds of 2023-2024 in the IGRA collection
The Diamonds of 2023-2024 in the IGRA collectionThe Diamonds of 2023-2024 in the IGRA collection
The Diamonds of 2023-2024 in the IGRA collection
Israel Genealogy Research Association
 
MARY JANE WILSON, A “BOA MÃE” .
MARY JANE WILSON, A “BOA MÃE”           .MARY JANE WILSON, A “BOA MÃE”           .
MARY JANE WILSON, A “BOA MÃE” .
Colégio Santa Teresinha
 
PIMS Job Advertisement 2024.pdf Islamabad
PIMS Job Advertisement 2024.pdf IslamabadPIMS Job Advertisement 2024.pdf Islamabad
PIMS Job Advertisement 2024.pdf Islamabad
AyyanKhan40
 
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdfANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
Priyankaranawat4
 
Types of Herbal Cosmetics its standardization.
Types of Herbal Cosmetics its standardization.Types of Herbal Cosmetics its standardization.
Types of Herbal Cosmetics its standardization.
Ashokrao Mane college of Pharmacy Peth-Vadgaon
 
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective UpskillingYour Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Excellence Foundation for South Sudan
 
Introduction to AI for Nonprofits with Tapp Network
Introduction to AI for Nonprofits with Tapp NetworkIntroduction to AI for Nonprofits with Tapp Network
Introduction to AI for Nonprofits with Tapp Network
TechSoup
 

Recently uploaded (20)

C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx
C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptxC1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx
C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx
 
How to Add Chatter in the odoo 17 ERP Module
How to Add Chatter in the odoo 17 ERP ModuleHow to Add Chatter in the odoo 17 ERP Module
How to Add Chatter in the odoo 17 ERP Module
 
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
 
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptxChapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
 
A Survey of Techniques for Maximizing LLM Performance.pptx
A Survey of Techniques for Maximizing LLM Performance.pptxA Survey of Techniques for Maximizing LLM Performance.pptx
A Survey of Techniques for Maximizing LLM Performance.pptx
 
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
 
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRMHow to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
 
Film vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movieFilm vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movie
 
The basics of sentences session 6pptx.pptx
The basics of sentences session 6pptx.pptxThe basics of sentences session 6pptx.pptx
The basics of sentences session 6pptx.pptx
 
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
 
writing about opinions about Australia the movie
writing about opinions about Australia the moviewriting about opinions about Australia the movie
writing about opinions about Australia the movie
 
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityNatural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
 
Main Java[All of the Base Concepts}.docx
Main Java[All of the Base Concepts}.docxMain Java[All of the Base Concepts}.docx
Main Java[All of the Base Concepts}.docx
 
The Diamonds of 2023-2024 in the IGRA collection
The Diamonds of 2023-2024 in the IGRA collectionThe Diamonds of 2023-2024 in the IGRA collection
The Diamonds of 2023-2024 in the IGRA collection
 
MARY JANE WILSON, A “BOA MÃE” .
MARY JANE WILSON, A “BOA MÃE”           .MARY JANE WILSON, A “BOA MÃE”           .
MARY JANE WILSON, A “BOA MÃE” .
 
PIMS Job Advertisement 2024.pdf Islamabad
PIMS Job Advertisement 2024.pdf IslamabadPIMS Job Advertisement 2024.pdf Islamabad
PIMS Job Advertisement 2024.pdf Islamabad
 
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdfANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
 
Types of Herbal Cosmetics its standardization.
Types of Herbal Cosmetics its standardization.Types of Herbal Cosmetics its standardization.
Types of Herbal Cosmetics its standardization.
 
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective UpskillingYour Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective Upskilling
 
Introduction to AI for Nonprofits with Tapp Network
Introduction to AI for Nonprofits with Tapp NetworkIntroduction to AI for Nonprofits with Tapp Network
Introduction to AI for Nonprofits with Tapp Network
 

If D is dense in a topological space (X, tau), prove that D is dense .pdf

  • 1. If D is dense in a topological space (X, tau), prove that D is dense in every topology smaller than tau. Solution If (X,T) is atopological space, then X is the set and T is the topology on X.T is also the family of subsets of X. If an element in the set X, D refers to 'dense', then, an element D in T refers to dense only. A subset can derive elements only from its superset.